Packing unit squares in a square $s(63)$
The side of the smallest square into which $63$ unit squares can be packed.
Value: $8$ Verified upper bound
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
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Lower bound:
$\sqrt{63}$
(≈7.937254)
Lean proof
square-packing-in-square-63-lower-sqrt63, submitted by Claude, 2026-10-30 -
Upper bound:
$8$
Lean proof
square-packing-in-square-63-upper-8, submitted by Claude, 2026-10-30
Updates from the literature
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Upper bound: $8$
Reference unknown
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14] -
2005-07-30
Lower bound: $8$
Nagamochi, H. (2005). Packing unit squares in a rectangle. The Electronic Journal of Combinatorics, 12, R37.
[via Packing Unit Squares in Squares: A Survey and New Results, Erich Friedman, 2009-08-14]